VNU Journal of Science, Mathematics - Physics 28 (2012) 94-97
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Concerning semi-quotient mappings
Nguyen Xuan Thuy
*D7- Thuy Ung - Hoa Binh – Thuong Tin – Hanoi, Vietnam Received 02 April 2012, received in revised from 14 June 2012
Abstract. In 1989, N.V. Velichko [1] introduced a semi-quotient ws-mapping, and proved that a sequential space has a point-countable k-network if and only if it is a semi-quotient ws-image of a metric space. Recently, Shou Lin and Jinjin Li [2] introduced and studied the concept of wks- mappings, wcs-mappings, and proved that every sequential space with a point-countable k-network is preserved by a continuous closed mapping. In this article, we introduce a class of mappings named wscc-mappings and give some properties of semi-quotient wscc-mappings. Moreover, we also give a result stating that every sequential space with a point-countable k-network is preserved by a continuous closed compact mapping.
Keywords: semi-quotient ws-mappings; wks-mappings; wcs-mappings; wscc-mappings; semi- quotient wscc-mappings
1. Introduction∗∗∗∗
A study of images of topological spaces under certain semi-quotient mappings is an important question in general topology. In 2009, to characterize spaces with a point-countable k-network as images of metric spaces under “nice” mappings, Shou Lin and Jinjin Li introduced concepts of wk- mappings, wc-mappings, wks-mappings, wcs-mappings in order to modify semi-quotient mappings. In this article, we introduce a class of mappings named wscc-mappings and give some properties of semi- quotient wscc-mappings.
Throughout this article, all spaces are assumed to be Hausdorff, all mappings are assumed onto.
For terms are not defined here, please refer to [3].
Definition 1.1 [1]. Suppose that a mapping f: X ⟶ Y, and is a subspace of X. the mapping f is called continuous about if for each x X and any neighborhood V of f(x) in Y there is a neighborhood W of x in X such that f(W ) V.
Denote = f : Y.
Lemma 1.2 [2]. Suppose that a mapping f: X ⟶ Y, and is a subspace of X. The following are equivalent:
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N.X. Thuy / VNU Journal of Science, Mathematics-Physics 28 (2012) 94-97 95
(1) f is continuous about .
If a net in converges to a point x in X, then a net converges to f(x) in Y.
(2) If T is a subset of Y, then ⊂ ( ).
Remark 1.3 [2]. By Lemma 1.2, the restriction f : Y is continuous f is continuous about ⟹ the striction = f : Y is continuous.
Definition 1.4 [1]. A mapping f : (X, ) ⟶ Y is called a semi-quotient ws-mapping if ⊂ X and the following are satisfied:
(1) The restriction = f : Y is an s-mapping, i. e., (y) is a separable subspace of for each y Y.
(2) f is continuous about .
(3) A subset T of Y is closed if and only if ⊂ (T).
Definition 1.5 [2]
(1) f: (X, ⟶ Y is called a ws-mapping if it satisfies the conditions (1) and (2) in Definition 1.4.
(2) f : (X, ⟶ Y is called a semi-quotient mapping if it satisfies the condition (3) in Definition 1.4.
Definition 1.6 [2]. Suppose that a mapping f : X ⟶ Y is continuous about .
(1) f: (X, ⟶ Y is called a wk-mapping if K is a compact subset of Y and T is a sequence in K, there is a sequence S in such that S has an accumulation in X and f(S) is a subsequence of T.
(2) f: (X, ⟶ Y is called a wc-mapping if T is a convergent sequence in Y, there is a sequence S in such that S has an accumulation in X and f(S) is a subsequence of T.
(3) f: (X, ⟶ Y is called a wks-mapping (wcs-mapping) if it is a wk-mapping (wc-mapping) and a ws-mapping.
Definition 1.7 [2],[4]. Suppose that f : X ⟶ Y is a continuous mapping.
(1) f is called a compact-covering mapping if K is a compact subset of Y, there is a compact subset L of X with f(L) = K.
(2) f is called a sequence-covering mapping if T is a convergent sequence including the limit point in Y, there is a compact subset L in X with f(L) = T.
Definition 1.8. Suppose that a mapping f : X ⟶ Y is continuous about . Then, f: (X, ⟶ Y is called a wscc-mapping if it is a compact-covering mapping and a ws-mapping.
Remark 1.9. The following statements hold.
(1) Compact-covering mappings ⟹ sequence-covering mappings [2].
(2) Compact-covering mappings ⟹ wk-mappings ⟹ wc-mappings [2].
(3) wscc-mappings ⟹ wks-mappings ⟹ wcs-mappings.
(4) wscc-mappings ⟹ sequence-covering ws-mappings.
Definition 1.10 [5]. A mapping f : X ⟶ Y is called weakly continuous if (V) [ ( )] for each open set V in Y. f : X ⟶ Y is weakly continuous if and only if for each x X and any neighborhood V of f(x) in Y, there is a neighborhood W of x in X with f(W) ⊂ .
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2. Main results
Theorem 2.1. Every continuous closed compact mapping is a semi-quotient wscc-mapping.
Proof. Suppose that f: X ⟶ Y is a continuous closed compact mapping. For a compact subset K of a space Y, and we put L = (K). Since f is closed compact mapping, L is a compact subset of X. This implies that there is a compact subset L of X with f(L) = f( (K)) = K. Therefore, f is a compact- covering mapping. On the other hand, for each y Y take an (y), and put = { : y Y}. It is obvious that, f : (X, ⟶ Y is continuous about and is a ws-mapping. If T is a subset of Y, and
⊂ (T), then = = f( ) T, thus T is closed in Y. Therefore f is a semi-quotient mapping. This implies that f is a semi-quotient wscc-mapping. The proof is complete.
Remark 2.2. From Theorem 2.1 and Remark 1.9, the following holds.
(1) Continuous closed compact mappings ⟹ semi-quotient wscc-mappings ⟹ semi-quotient wks- mappings ⟹ semi-quotient wcs-mappings.
(2) Semi-quotient wscc-mappings ⟹ semi-quotient sequence-covering ws-mappings.
Theorem 2.3. Let f: X ⟶ Y be a continuous closed compact mapping, and X be a sequential space, be a subspace of a metric space M. If g: (M, ⟶ X is a weakly continuous semi-quotient wscc-mapping. Then the composition h = g is a semi-quotient wscc-mapping.
First, let us prove a lemma.
Lemma 2.4. If f : X ⟶ Y and g : Y ⟶ Z are compact-covering mappings, then h = f is also a compact-covering mapping.
Proof. Suppose that K is a compact subset of Z, because g is compact-covering mapping, there exists a compact subset of Y such that g ( ) = K. On the other hand, since is a compact subset in Y and f is compact-covering mapping, there is a compact subset L in X with f(L) = . Therefore, there exists a compact subset L in X such that h (L) = ( f) (L) = g(f(L)) = g( ) = K. This shows that h is a compact-covering mapping.
Now, we give a proof of Theorem 2.3.
Firstly, for each y Y, take an (y), and put = ({ : y Y}), h = g: M ⟶ Y and = h . Since , h : (M, ⟶ Y is a ws-mapping and continuous about Now, we show that h is a semi-quotient mapping. Suppose that T is a non-closed subset of Y, thus there is a sequence { } in T such that the sequence { } converges to y T in Y. For each n , put = and let ॒ = { : n }. Since f is closed, ॒ is not closed in X, and since X is a sequential space, the sequence { } has a convergent subsequence. We can assume that the sequence { } converges to a point x in X, then f(x) = y. Because g is semi-quotient and ॒ is not closed in X, there exists a m \ (॒). We shall prove that g(m) = x. Indeed, if g(m) x, then there is a neighborhood V of g(m) in X such that = . Since g is weakly continuous, there exists a neighborhood W of m in M such that g(W) , thus W (॒) = , and it implies that m
. This contradicts to m \ (॒). Therefore, g(m) = x and h(m) = y T. For each open neighborhood U of m in M, U (T) U (॒) = U (॒) thus m \ (T), hence (T). It implies that h is a semi-quotient mapping. On the other hand, in view of the proof of Theorem 2.1, f is a compact-covering mapping. Finally, because g
N.X. Thuy / VNU Journal of Science, Mathematics-Physics 28 (2012) 94-97 97
is also a compact-covering mapping, by Lemma 2.4, h is a compact-covering mapping. Therefore, h is a semi-quotient wscc-mapping and so completes the proof.
Remark 2.5. In Theorem 2.3, if sequential space X is a sequential space with a point-countable k- network, then, because the closed mappings are quotient mappings and sequential spaces are preserved by quotient mappings [6], Y is a sequential space. By Corollary 16 in [2] and Theorem 2.3, we have the following corollary.
Corollary 2.6. Every sequential space with a point-countable k-network is preserved by a continuous closed compact mapping.
3. Conclusion
In this article, a class of mappings, called wscc-mappings is introduced. Besides, some theorems are obtained, which improve some results of Shou Lin and Jinjin Li.
Acknowledgment
The author would like to thank the referee for many valuable comments.
References
[1] N.V. Velichko, Ultrasequential spaces, Mat. Zametki, 45(2) (1989), 15-21 (in Russian) (=Math. Notes, 45, (1989), 99-103.
[2] Shou Lin, Generalized Metric Spaces and Mappings, Second Edition, Beijing: Science Press, 2007 (in Chinese).
[3] G. Gruenhage, E. Michael, and Y. Tanaka, Spaces determined by point- countable covers, Pacific J. Math., 113, (1984), 303-332.
[4] Shou Lin, Jinjin Li, Semi-quotient mappings and spaces with compact-countable k-networks, Advances in Mathematics(China), 38(4), (2009), 417-426.
[5] R. Engelking, General Topology, Berlin: Heldermann, (1989).
[6] N. Levine, A decomposition of continuity in topological spaces, Amer. Math. Monthly, 68, (1961), 44-46.
